BUG: `Series.std` and `Series.var` give incorrect results for complex values.

[randolf-scholz]

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Reproducible Example

import numpy as np
import pandas as pd

arr = np.array([-1j, 0j, 1j], dtype=complex)
s = pd.Series(arr, dtype=complex)

print(arr.std(ddof=0))  # 0.816496580927726
print(s.std(ddof=0))  # nan
print(arr.var(ddof=0))  # 0.666
print(s.var(ddof=0))  # -0.666

Issue Description

1. The results diverge from numpy.
2. pandas yields nonsensical results like negative floats.

Expected Behavior

For complex variables, std and var should give non-negative floating results. Recall that $σ ≔ \sqrt{𝔼|x-μ|^2 }$. Often, authors that only use real-valued variables leave out the absolute value, which I guess is what happened here.

Installed Versions

INSTALLED VERSIONS

commit : 2cc3762
python : 3.13.4
python-bits : 64
OS : Linux
OS-release : 6.11.0-26-generic
Version : #26~24.04.1-Ubuntu SMP PREEMPT_DYNAMIC Thu Apr 17 19:20:47 UTC 2
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 2.3.0
numpy : 2.3.0
pytz : 2025.2
dateutil : 2.9.0.post0
pip : 25.1.1
Cython : None
sphinx : 8.2.3
IPython : 9.3.0
adbc-driver-postgresql: None
adbc-driver-sqlite : None
bs4 : 4.13.4
blosc : None
bottleneck : None
dataframe-api-compat : None
fastparquet : None
fsspec : 2025.5.1
html5lib : None
hypothesis : 6.135.9
gcsfs : None
jinja2 : 3.1.6
lxml.etree : None
matplotlib : 3.10.3
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
psycopg2 : None
pymysql : None
pyarrow : 20.0.0
pyreadstat : None
pytest : 8.4.0
python-calamine : None
pyxlsb : None
s3fs : None
scipy : 1.15.3
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
xlsxwriter : None
zstandard : None
tzdata : 2025.2
qtpy : None
pyqt5 : None